Polytope of Type {2,110,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,110,4}*1760
if this polytope has a name.
Group : SmallGroup(1760,1253)
Rank : 4
Schlafli Type : {2,110,4}
Number of vertices, edges, etc : 2, 110, 220, 4
Order of s₀s₁s₂s₃ : 220
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,110,2}*880
   4-fold quotients : {2,55,2}*440
   5-fold quotients : {2,22,4}*352
   10-fold quotients : {2,22,2}*176
   11-fold quotients : {2,10,4}*160
   20-fold quotients : {2,11,2}*88
   22-fold quotients : {2,10,2}*80
   44-fold quotients : {2,5,2}*40
   55-fold quotients : {2,2,4}*32
   110-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4, 13)(  5, 12)(  6, 11)(  7, 10)(  8,  9)( 14, 47)( 15, 57)( 16, 56)
( 17, 55)( 18, 54)( 19, 53)( 20, 52)( 21, 51)( 22, 50)( 23, 49)( 24, 48)
( 25, 36)( 26, 46)( 27, 45)( 28, 44)( 29, 43)( 30, 42)( 31, 41)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 59, 68)( 60, 67)( 61, 66)( 62, 65)( 63, 64)
( 69,102)( 70,112)( 71,111)( 72,110)( 73,109)( 74,108)( 75,107)( 76,106)
( 77,105)( 78,104)( 79,103)( 80, 91)( 81,101)( 82,100)( 83, 99)( 84, 98)
( 85, 97)( 86, 96)( 87, 95)( 88, 94)( 89, 93)( 90, 92)(114,123)(115,122)
(116,121)(117,120)(118,119)(124,157)(125,167)(126,166)(127,165)(128,164)
(129,163)(130,162)(131,161)(132,160)(133,159)(134,158)(135,146)(136,156)
(137,155)(138,154)(139,153)(140,152)(141,151)(142,150)(143,149)(144,148)
(145,147)(169,178)(170,177)(171,176)(172,175)(173,174)(179,212)(180,222)
(181,221)(182,220)(183,219)(184,218)(185,217)(186,216)(187,215)(188,214)
(189,213)(190,201)(191,211)(192,210)(193,209)(194,208)(195,207)(196,206)
(197,205)(198,204)(199,203)(200,202);;
s2 := (  3, 15)(  4, 14)(  5, 24)(  6, 23)(  7, 22)(  8, 21)(  9, 20)( 10, 19)
( 11, 18)( 12, 17)( 13, 16)( 25, 48)( 26, 47)( 27, 57)( 28, 56)( 29, 55)
( 30, 54)( 31, 53)( 32, 52)( 33, 51)( 34, 50)( 35, 49)( 36, 37)( 38, 46)
( 39, 45)( 40, 44)( 41, 43)( 58, 70)( 59, 69)( 60, 79)( 61, 78)( 62, 77)
( 63, 76)( 64, 75)( 65, 74)( 66, 73)( 67, 72)( 68, 71)( 80,103)( 81,102)
( 82,112)( 83,111)( 84,110)( 85,109)( 86,108)( 87,107)( 88,106)( 89,105)
( 90,104)( 91, 92)( 93,101)( 94,100)( 95, 99)( 96, 98)(113,180)(114,179)
(115,189)(116,188)(117,187)(118,186)(119,185)(120,184)(121,183)(122,182)
(123,181)(124,169)(125,168)(126,178)(127,177)(128,176)(129,175)(130,174)
(131,173)(132,172)(133,171)(134,170)(135,213)(136,212)(137,222)(138,221)
(139,220)(140,219)(141,218)(142,217)(143,216)(144,215)(145,214)(146,202)
(147,201)(148,211)(149,210)(150,209)(151,208)(152,207)(153,206)(154,205)
(155,204)(156,203)(157,191)(158,190)(159,200)(160,199)(161,198)(162,197)
(163,196)(164,195)(165,194)(166,193)(167,192);;
s3 := (  3,113)(  4,114)(  5,115)(  6,116)(  7,117)(  8,118)(  9,119)( 10,120)
( 11,121)( 12,122)( 13,123)( 14,124)( 15,125)( 16,126)( 17,127)( 18,128)
( 19,129)( 20,130)( 21,131)( 22,132)( 23,133)( 24,134)( 25,135)( 26,136)
( 27,137)( 28,138)( 29,139)( 30,140)( 31,141)( 32,142)( 33,143)( 34,144)
( 35,145)( 36,146)( 37,147)( 38,148)( 39,149)( 40,150)( 41,151)( 42,152)
( 43,153)( 44,154)( 45,155)( 46,156)( 47,157)( 48,158)( 49,159)( 50,160)
( 51,161)( 52,162)( 53,163)( 54,164)( 55,165)( 56,166)( 57,167)( 58,168)
( 59,169)( 60,170)( 61,171)( 62,172)( 63,173)( 64,174)( 65,175)( 66,176)
( 67,177)( 68,178)( 69,179)( 70,180)( 71,181)( 72,182)( 73,183)( 74,184)
( 75,185)( 76,186)( 77,187)( 78,188)( 79,189)( 80,190)( 81,191)( 82,192)
( 83,193)( 84,194)( 85,195)( 86,196)( 87,197)( 88,198)( 89,199)( 90,200)
( 91,201)( 92,202)( 93,203)( 94,204)( 95,205)( 96,206)( 97,207)( 98,208)
( 99,209)(100,210)(101,211)(102,212)(103,213)(104,214)(105,215)(106,216)
(107,217)(108,218)(109,219)(110,220)(111,221)(112,222);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(222)!(1,2);
s1 := Sym(222)!(  4, 13)(  5, 12)(  6, 11)(  7, 10)(  8,  9)( 14, 47)( 15, 57)
( 16, 56)( 17, 55)( 18, 54)( 19, 53)( 20, 52)( 21, 51)( 22, 50)( 23, 49)
( 24, 48)( 25, 36)( 26, 46)( 27, 45)( 28, 44)( 29, 43)( 30, 42)( 31, 41)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 59, 68)( 60, 67)( 61, 66)( 62, 65)
( 63, 64)( 69,102)( 70,112)( 71,111)( 72,110)( 73,109)( 74,108)( 75,107)
( 76,106)( 77,105)( 78,104)( 79,103)( 80, 91)( 81,101)( 82,100)( 83, 99)
( 84, 98)( 85, 97)( 86, 96)( 87, 95)( 88, 94)( 89, 93)( 90, 92)(114,123)
(115,122)(116,121)(117,120)(118,119)(124,157)(125,167)(126,166)(127,165)
(128,164)(129,163)(130,162)(131,161)(132,160)(133,159)(134,158)(135,146)
(136,156)(137,155)(138,154)(139,153)(140,152)(141,151)(142,150)(143,149)
(144,148)(145,147)(169,178)(170,177)(171,176)(172,175)(173,174)(179,212)
(180,222)(181,221)(182,220)(183,219)(184,218)(185,217)(186,216)(187,215)
(188,214)(189,213)(190,201)(191,211)(192,210)(193,209)(194,208)(195,207)
(196,206)(197,205)(198,204)(199,203)(200,202);
s2 := Sym(222)!(  3, 15)(  4, 14)(  5, 24)(  6, 23)(  7, 22)(  8, 21)(  9, 20)
( 10, 19)( 11, 18)( 12, 17)( 13, 16)( 25, 48)( 26, 47)( 27, 57)( 28, 56)
( 29, 55)( 30, 54)( 31, 53)( 32, 52)( 33, 51)( 34, 50)( 35, 49)( 36, 37)
( 38, 46)( 39, 45)( 40, 44)( 41, 43)( 58, 70)( 59, 69)( 60, 79)( 61, 78)
( 62, 77)( 63, 76)( 64, 75)( 65, 74)( 66, 73)( 67, 72)( 68, 71)( 80,103)
( 81,102)( 82,112)( 83,111)( 84,110)( 85,109)( 86,108)( 87,107)( 88,106)
( 89,105)( 90,104)( 91, 92)( 93,101)( 94,100)( 95, 99)( 96, 98)(113,180)
(114,179)(115,189)(116,188)(117,187)(118,186)(119,185)(120,184)(121,183)
(122,182)(123,181)(124,169)(125,168)(126,178)(127,177)(128,176)(129,175)
(130,174)(131,173)(132,172)(133,171)(134,170)(135,213)(136,212)(137,222)
(138,221)(139,220)(140,219)(141,218)(142,217)(143,216)(144,215)(145,214)
(146,202)(147,201)(148,211)(149,210)(150,209)(151,208)(152,207)(153,206)
(154,205)(155,204)(156,203)(157,191)(158,190)(159,200)(160,199)(161,198)
(162,197)(163,196)(164,195)(165,194)(166,193)(167,192);
s3 := Sym(222)!(  3,113)(  4,114)(  5,115)(  6,116)(  7,117)(  8,118)(  9,119)
( 10,120)( 11,121)( 12,122)( 13,123)( 14,124)( 15,125)( 16,126)( 17,127)
( 18,128)( 19,129)( 20,130)( 21,131)( 22,132)( 23,133)( 24,134)( 25,135)
( 26,136)( 27,137)( 28,138)( 29,139)( 30,140)( 31,141)( 32,142)( 33,143)
( 34,144)( 35,145)( 36,146)( 37,147)( 38,148)( 39,149)( 40,150)( 41,151)
( 42,152)( 43,153)( 44,154)( 45,155)( 46,156)( 47,157)( 48,158)( 49,159)
( 50,160)( 51,161)( 52,162)( 53,163)( 54,164)( 55,165)( 56,166)( 57,167)
( 58,168)( 59,169)( 60,170)( 61,171)( 62,172)( 63,173)( 64,174)( 65,175)
( 66,176)( 67,177)( 68,178)( 69,179)( 70,180)( 71,181)( 72,182)( 73,183)
( 74,184)( 75,185)( 76,186)( 77,187)( 78,188)( 79,189)( 80,190)( 81,191)
( 82,192)( 83,193)( 84,194)( 85,195)( 86,196)( 87,197)( 88,198)( 89,199)
( 90,200)( 91,201)( 92,202)( 93,203)( 94,204)( 95,205)( 96,206)( 97,207)
( 98,208)( 99,209)(100,210)(101,211)(102,212)(103,213)(104,214)(105,215)
(106,216)(107,217)(108,218)(109,219)(110,220)(111,221)(112,222);
poly := sub<Sym(222)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope